Edge Universality for Inhomogeneous Random Matrices
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866917072594796544 |
|---|---|
| author | Liu, Dang-Zheng Zou, Guangyi |
| author_facet | Liu, Dang-Zheng Zou, Guangyi |
| contents | We consider symmetric and Hermitian random matrices whose entries are independent and symmetric random variables with an arbitrary variance pattern. Under a novel Short-to-Long Mixing condition, which is sharp in the sense that it precludes a corrected shift at the spectral edge, we establish GOE/GUE edge universality for such inhomogeneous random matrices. This condition effectively reduces the universality problem to verifying the mixing properties of a random walk governed by the variance profile matrix.
Our universality results are applicable to a remarkably broad class of random matrix ensembles that may be highly inhomogeneous, sparse, or far beyond the mean-field setting of classical random matrix theory. Notable examples include:
1. Inhomogeneous Wishart-type random matrices;
2. Random band matrices whose entries are independent random variables with general variance profile, particularly with an optimal bandwidth in dimensions $d \le 2$;
3. Sparse random matrices with structured variance profiles;
4. Generalized Wigner matrices under significantly weaker sparsity constraints and heavy-tailed entry distributions;
5. Wegner orbital models under sharp mixing assumptions;
6. Random 2-lifts of random $d$-regular graphs where $d\geq N^{2/3+ε}$ for any $ε>0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_17838 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Edge Universality for Inhomogeneous Random Matrices Liu, Dang-Zheng Zou, Guangyi Probability Mathematical Physics Spectral Theory 60B20, 60F05 We consider symmetric and Hermitian random matrices whose entries are independent and symmetric random variables with an arbitrary variance pattern. Under a novel Short-to-Long Mixing condition, which is sharp in the sense that it precludes a corrected shift at the spectral edge, we establish GOE/GUE edge universality for such inhomogeneous random matrices. This condition effectively reduces the universality problem to verifying the mixing properties of a random walk governed by the variance profile matrix. Our universality results are applicable to a remarkably broad class of random matrix ensembles that may be highly inhomogeneous, sparse, or far beyond the mean-field setting of classical random matrix theory. Notable examples include: 1. Inhomogeneous Wishart-type random matrices; 2. Random band matrices whose entries are independent random variables with general variance profile, particularly with an optimal bandwidth in dimensions $d \le 2$; 3. Sparse random matrices with structured variance profiles; 4. Generalized Wigner matrices under significantly weaker sparsity constraints and heavy-tailed entry distributions; 5. Wegner orbital models under sharp mixing assumptions; 6. Random 2-lifts of random $d$-regular graphs where $d\geq N^{2/3+ε}$ for any $ε>0$. |
| title | Edge Universality for Inhomogeneous Random Matrices |
| topic | Probability Mathematical Physics Spectral Theory 60B20, 60F05 |
| url | https://arxiv.org/abs/2508.17838 |